My alternator and battery both bad after just 1 year.
Mechanic claims the alternator was fried out because of a defective battery – bad cell? Can this be true?

Yes! A battery that has a bad cell runs at a voltage considerably lower than the normal 13V - 14V with the engine running and the alternator charging. Lots of modern alternators do not use permanent magnets, but a field coil to provide magnetism to the alternator. That field coil (an electromagnet) takes voltage from the battery to create magnetism, therefore making the alternator charge. As with anything else electrical, as the voltage drops (bad cell in the battery) the current goes up, and it's the higher current, over time, that can kill the alternator.

By the way, another thing or fifty(50) can happen: These two happened to me.

New battery, new alternator, old car: Accidentally drained, left the lights on, say. After the jump start, the new hungry battery and the new potent alternator may melt your old dash ammeter. Especially MoPar: so proud of introducing the alternator to replace the generator, they insisted on having a dashboard ammeter thru at least 1967. Workaround: stop the engine after five minutes and let all cool for five minutes. Jump again if needed, but probably won't be. Repeat as needed until less than 20 amps are flowing.

New battery, weak, car in shop: The "mechanic" (?) slaps on the 40 or 60 amp quick charger and DOES NOT loosen the cell caps. The gases generated blow a hole in a cell wall and your 6 cell battery is now a 5 cell. Workaround: Explain that you insist on watching and say or write a liability release for your presence in shop; else leave. { Move car out of shop, call taxi, tip $10 for a jump from the taxi. Works only if the taxi driver is over 30.}

this is not true.

the alternator is designed to charge the battery. a low batery will NOT damage an alternator.

i have been an ASE master for more than 30 years and I have never seen a battery damage an alternator, as long as the battery is installed  properly.

that sucks man

  yes, that is why you need to have these things checked frequently, too much powe  or defective battery can really damage your alternator. this happened to my car.

 well i think it is. the battery might have damaged the alternator but not to the point that it gets fried. there must be wrong with the wiring and that might be the reason for the burnt alternator.